Next, we connect our results pointwise using range functions (Theorem REF  and REF ). The range functions are generally employed for the analysis of the local behavior of the MI spaces started by Helson [1]}. For the setting of MI spaces, the global and local behavior of a frame agrees well with each other. Bownik et.al. in paper [2]} proved
that the system generated by multiplications is a frame (Bessel) if and only if the corresponding local system is a frame (Bessel) almost everywhere. The MI operators has a one-to-one correspondence with range operators acting locally over fibers indexed by \(x \in X\) . Furthermore, many properties such as unitary equivalence of the MI systems, dual frame, and orthogonality for a pair of frames are also preserved. The reader can refer to [3]}, [4]} for more details on range functions.
